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2 Dynamical Systems AnalysisWhy? Often have sets of differential equations describing a dynamical system.Analysis determines how the system behaves over time, in particular investigating future behaviour of the system given any current state ie the long-term behaviour of the system. Can solve equations for particular starting point(s), but this is often not enough to enable us to understand the systemTherefore use complementary analysis focused on finding equilibrium states (or stationary/critical/fixed points) where system remains unchanged over timeAlso try to classify these states/points as stable/unstable by investigating the behaviour of the system near them

3 Eg coin balanced on a table: how many equilibria?3: Heads, Tails, EdgeStart on its edge, it is in equilibrium … is it stable?… No! Small movement away (perturbation) means that coin will end up in one of 2 different equilibria: Heads/ Tails.Both stable as perturbation does not result in a change in stateGiven a noisy world, coin will end up in a stable stateNote however, may not be able to tell which state it will end up in … idea of chaotic systemsSimilarly, think of a ball at rest in a dark landscape. It’s either on top of a hill or at the bottom of a valley. To find out which, push it (perturb it), and see if it comes back. (What about flat bits?)

4 By the end you will be able to:Find fixed points of a systemClassify fixed points as stable/unstable etcUse graphical methods (direction and cobweb plots) to analyse the behaviour of systemsUse phase-plane analysis to analyse the behaviour of systems1st half: 1 (space) dimensional systems2nd half: 2 (and higher) dimensional systemsWill NOT talk about:Proofs of many of the stability theorems – focus will be on how to use them to analyse your systemsThe many special cases – these lectures are a primer

9 x=0 unstable and x=1 stable. What if we want more info?What happens if we don’t start from a fixed point? Iterate the system to get behaviourWhat happens if we start at any starting positions eg x(0) =0.8 or any of the points shown by blue dots?Use direction fields. Idea is to plot x against t and, at a grid of points on the plane, draw an arrow indicating the direction of subsequent points x(t+h), thus showing the movement of x for different starting conditions.

10 Direction fieldsWhat is this direction? remember: x(t+h) = x(t) + h dx/dtDrawing an arrow from x(t) to x(t+h), shows the direction of next point so arrow starts at (t, x(t)) and goes to (t+h, x(t)+hdx/dt)Matlab code to generate direction : use ‘quiver’[T,X]=meshgrid([0:0.1:1.4], [0:0.1:0.9]);NewT= ones(size(T));NewX=6*X.*(1-X);quiver(T,X,NewT,NewX)Don’t need h as it is scaled automatically

11 Can use them to eg see the influence of parameters on systems6x(1-x)3x(1-x)Rise is less steep-6x(1-x)0x(1-x)All points stable: degenerate solutionNow 0 is stable and 1 unstable

12 Recap: Have looked at analysis of 1D systemsAnalysis focussed on finding fixed points and analysing stabilityShowed that fixed points could be found by solving:dx/dt = 0Next we will examine discrete dynamical systems then systems with more dimensions!Introduce cobweb plots as an analytical aid

13 Discrete dynamical systemsNow widen discussion to discrete dynamical systems where:x(n+1) = f(x(n))for a sequence of time steps, t0,t1, t2, ..., tn, tn+1, ...ie x(n) is shorthand for “the value of x at the n’th timestep tn”Fixed points are those where x(n+1) = x(n) so find them by solving: a = f(a)[NB this is same as if dx/dt=0 sinceDefine f(x(n)) = x(n) + hdx/dt,then if: dx/dt = 0 , x(n+1) = f(x(n)) = x(n) + 0 = x(n)]However, the dependency on the step-size h is now explicitImportant as stability can be determined by value of h used

15 However, these plots can also tell us about the stability of the fixed points via the following procedure:Start at a point x(0)=0.2xx(0)

16 Go vertically up to the curve f(x)Start at a point x(0)=0.2Go vertically up to the curve f(x)This is f(x(0))= x(1)y=f(x(0))= x(1)xx(0)

17 Go horizontally from this point to the line y=x as this is x(1) on the x-axis.y=f(x(0))= x(1)xx(1)

18 Get x(2) from going vertically from this point to f(x)Go horizontally from this point to the line y=x as this is x(1) on the x-axis.Get x(2) from going vertically from this point to f(x)This is f(x(1))= x(2)y=f(x(1))= x(2)xx(1)

19 Then get x(3) from going vertically from this point to f(x) since:Go horizontally from this point to the line y=x to get x(2) on the x-axisThen get x(3) from going vertically from this point to f(x) since:f(x(2)) = x(3)y=f(x(2))= x(3)x(3)x

20 x(6) x(5) Then get x(4), x(5) etc etc x(4)Can therefore classify stability of fixed points by seeing which starting points lead to which fixed pointx(4)x

22 Semi-Stable from below y = x3 – x2 + 1x=1.2 stable, x=1.4 unstable.y = 0.8x - x3xxLast eg illustrates the concept of a basin of attraction: the range of values of a that will lead to a stable point if started from (or if passed through)

23 Leads us to following theorem ...Notice that if the gradient of y=f(x) evaluated at a0 (the fixed point) is > gradient of y=x (which equals 1) the point is unstabley = 0.6x + 0.2y = 2x - 0.5xxStableUnstableLeads us to following theorem ...

29 Summary So far: how to find fixed points of a dynamical systemconcept of stability and its dependence on parametersDirection fields for determining behaviourcobweb plots for stabilityhow to check the stability of a fixed pointIn seminars:introduce a few more examplesshow how the wrong choice of a time-step leads to instabilitywork through an example of this analysis used for GasNetsNow2D (and higher) systems...

30 2 and higher dimensional systemsIn higher-dimensional systems movement of trajectories can exhibit a wider range of dynamical behaviourFixed points still exist, but can be more interesting depending on how trajectories approach or repel from the equilibrium point eg system could spiral in to a stable pointAlso, other types of stability exist eg saddle-nodes, and importantly cyclic/periodic behaviour: limit cyclesMore interesting, but more difficult to analyse… We will cover:How to find fixed pointsClassifying fixed points for linear systemsPhase-plane (phase-space) analysis of behaviour of system

31 2D systemsAnalyse 2D (and multi-D) systems in a similar way to 1d systems.Won’t go into proofs (see eg introduction to ordinary differential equations, Saperstone and refs at end) but will give general procedureSuppose we have the following systemFind fixed pointsExamine stability of fixed pointsExamine the phase plane and isoclines/trajectories

33 Examine behaviour at/near fixed points but view in phase-space (or phase-plane) where x plotted against y rather than against timeeg predator-prey from last week: phase plane gives extra infox and y over timex and y in phase-planeCyclic behaviour is fixed but system not at a fixed point: complications of higher dimensionsNeed phase plane AND fixed points to analyse behaviour: [DEMO]

34 Classify Fixed PointsSuppose x0 =(x0, y0)T is a fixed point. Define the Jacobian:whereandFind eigenvalues and eigenvectors of J evaluated at the fixed point:If eigenvalues have negative real parts, x0 asymptotically stableIf at least one has positive real part, x0 unstableIf eigenvalues are pure imaginary, stable or unstableComplex numbers are made up of a real part (normal number) and an imaginary part eg 4 + 3i where

35 ie Repeatedly transform v by A.Remember from matrix lecture: Intuition is direction of x is unchanged by being transformed by A so it reflects the principal direction (or axis) of the transformationxxie Repeatedly transform v by A.xxStart at vthen Avthen AAv= A2vetc …xMost starting points result in curved trajectoriesStarting from an eigenvector, x however, get:xAx = lx,A2x = l2x,A3x = l3x,A4x = l4x, …xxSo trajectory is a straight linexxNote if |l| > 1, x expands. If not, will contract

36 Fixed Points of Linear SystemsVarious behaviours depending on the eigenvalues (ei) and eigenvectors (vi) of JIn general, points attracted along negative eigenvalues and repelled by positive. Axes of attraction etc are eigenvectors egUnstable node,e1 > e2 > 0(Stable node is same but arrows pointing the other way)

39 Linear centre, complex ei’s, real part = 0For non-linear equations, behaviour near the fixed points will be ‘almost like’ the behaviour of a linear system depending how ‘almost linear’ it isBehaviour gets less linear-like the further away trajectories get from the fixed point

41 have a saddle node at (0,0) and a centre at (80, 12)yxxxTo get a more accurate picture, we can look at all the direction vectors in the phase plane

42 Phase plane analysisySimilar to direction fields except we use a plot of x against y(x(t+h),y(t+h))Want to examine behaviour of the dynamical system from different starting points(x(t),y(t))x1. Select a set of starting points (x(t),y(t))2. By Euler evaluate:(x(t+h),y(t+h))=(x(t)+hdx/dt, y(t) +hdy/dt)plot an arrow depicting the direction of movement

43 Isoclines Often helpful to plot isoclines on the phase planeIsoclines are curves where dx/dt or dy/dt are constantFound by setting dx/dt = c and dy/dt = c and solvingThe most useful are nullclines, where dx/dt or dy/dt = 0 since1. Points where the nullclines cross are fixed points2. Trajectories cross the nullclines at right angles so we know in which direction they are movingWhy? New point (x(t+h), y(t+h)) = (x(t)+hdx/dt, y(t)+hdy/dt)If: dx/dt = 0 x(t+h) = x(t) so movement is verticalIf: dy/dt = 0 y(t+h) = y(t) so movement is horizontal

44 can be more useful in a more complicated example eg Fitzhugh-Nagumow- nullclineV- nullclinefixed pointboth the components vanish

46 To see the behaviour of the system, use all the info gatheredxydx/dt=0dy/dt = 0stability analysis and isoclinesphase-planetrajectories in phase space and over timeand build up a picture of what will happen

47 Chaos and stability Higher dimensional systems are VERY often chaotic"No one [chaos scientist he interviewed] could quite agree on [a definition of] the word itself” (Gleik 1988, quoted in 11/08)My notion of chaos is unpredictability: things starting arbitrarily close together can get arbitrarily far apart (but do not necessarily do so)For real systems, means that finest-grain of detail of initial conditions is important ie the butterfly effectFor analysis, means it is a pain if not impossible unless constrained somehow, which is often the caseHave the concept of Lyapunov Stability, things getting arbitrarily close to a fixed point, but maybe not being there all the time, and looser notions

48 Limit CyclesCan also approach a set of points: consider a periodic attractor with period 3;What about period p?Limit cycles or orbits also become possible, where the state of the system ‘cycles’ through a set of states.Limit Cycle: “An attracting set to which orbits or trajectories converge and upon which trajectories are periodic.” (http://mathworld.wolfram.com/LimitCycle.html, 11/08)There are many different kinds of limit cycles eg periodic or chaotic, stable, unstable, or sometimes even half-stable (Strogatz, 1994).Limit cycles are important because eg they represent systems that oscillate even in the absence of external input.

49 One Man’s Transient is Another Man’s Limit Cycle?Bi-stableMono-stableFrom Buckley et al. (2008). Monostable controllers for adaptive behaviour. In Proc SAB 10‘Transient’: any point not in/at(/cycling round?) an equilibriumBuckley et al showed you can get bi-stability from a mono-stable systemShows the importance of transientsAlso an issue of ‘strict’ definitions? Both have 2 attractors … discuss

50 Multi-dimensional 1st order systemsIf system is more than 2 dimensional, can use the same techniquesie find Jacobian (will be an nxn matrix if we have n equations)However can only view 2 or 3 variables at once… and for non-linear gets pretty complicated pretty quickly but we can but try

51 Summary This lecture: how to find fixed points of a dynamical systemHow to analyse stability of fixed pointsthe use of the phase plane to determine the behaviourIn seminars:work through analysis of a GasNet neuronanalyse dynamics of 2 linked GasNet neurons (NB similar to analysis of CTRNN neurons)

52 RefsMany refs for dynamical systems. Some will be more suited to your level and you may find others after looking at these :General refs:Saperstone, (1998) Introduction to ordinary differential equationsGoldstein, H. (1980). Classical Mechanics.Strogatz, S. (1994) Nonlinear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry, and Engineering.Gleick, J. (1988) Chaos: Making a New Science.More on analysis and applicationsRosen, R. (1970). Dynamical System Theory in Biology, volume I. Stability theory and its applications.Rubinow, S. I. (1975). Introduction to Mathematical Biology.Good for discrete systems1. Sandefur, J. T. (1990). Discrete Dynamical Systems: Theory and Applications.

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